Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically It is measured in coulombs per metre (C/m), square metre (C/m²), or cubic metre (C/m³), respectively. The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International M^2 redirects here For other uses see M². CM2 redirects here CM3 redirects here If you were looking for the 3rd game in the Cooking Mama series abbreviated as CM3 see here. Since there are positive as well as negative charges, the charge density can take on negative values. Like any density it can depend on position. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different It should not be confused with the charge carrier density. The charge carrier density denotes the number of Charge carriers per Volume. As related to chemistry, it can refer to the charge distribution over the volume of a particle, molecule, or atom. Therefore, a lithium cation will carry a higher charge density than a sodium cation due to its smaller ionic radius.

1 Classical charge density
2 Quantum charge density
3 Application
4 See also
5 References

Classical charge density

Continuous charges

The integral of the charge density $\alpha_q(\mathbf r)$ , $\sigma_q(\mathbf r)$ , $\rho_q(\mathbf r)$ over a line $l$ , surface $S$ , or volume $V$ , is equal to the total charge $Q$ of that region, defined to be: ^[1]

$Q=\int\limits_L \alpha_q(\mathbf r) dl$ ,

$Q=\int\limits_S \sigma_q(\mathbf r) dS$ ,

$Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V.$

This relation defines the charge density mathematically. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Note that the symbols used to denote the various dimensions of charge density vary between fields of studies. Other commonly used notations are $λ$ , $σ$ , $ρ$ ; or $ρ l$ , $ρ s$ , $ρ v$ for (C/m), (C/m²), (C/m³) and respectively.

Homogeneous charge density

For the special case of a homogeneous charge density, that is one that is independent of position, equal to $ρ q,0$ the equation simplifies to:

$Q=V\cdot \rho_{q,0}$

The proof of this is simple. For other uses see Homogeneous. In Physics, homogeneous mixtures are mixtures that have definite consistent composition and properties Start with the definition of the charge of any volume:

$Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V$

Then, by definition of homogeneity, $\rho_q(\mathbf r)$ is a constant that we will denote $ρ q,0$ to differentiate between the constant and non-constant forms, and thus by the properties of an integral can be pulled outside of the integral resulting in:

$Q=\rho_{q,0} \int\limits_V \,\mathrm{d}V$

Again, by the properties of integrals:

$\int\limits_V \,\mathrm{d}V = V$

Therefore by substitution:

$\rho_{q,0} \int\limits_V \,\mathrm{d}V = V\cdot \rho_{q,0}$

Which leads to:

$Q=V\cdot \rho_{q,0}$

Which is precisely the result mentioned above for volume charge density. The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.

Discrete charges

If the charge in a region consists of $N$ discrete point-like charge carriers like electrons the charge density can be expressed via the Dirac delta function, for example, the volume charge density is: Here, and $\mathbf r_i$ the position of the $i$ th charge carrier. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. If all charge carriers have the same charge $q$ (for electrons $q = - e$ ) the charge density can be expressed through the charge carrier density $n(\mathbf r)$ : Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.

Quantum charge density

In quantum mechanics, charge density is related to wavefunction $\psi(\mathbf r)$ by the equation

$\rho_q(\mathbf r) = q\cdot|\psi(\mathbf r)|^2$

when the wavefunction is normalized as

$Q= q\cdot \int |\psi(\mathbf r)|^2 \, d\mathbf r$

Application

The charge density appears in the continuity equation which follows from Maxwell's Equations in the electromagnetic theory. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

References

^ Spacial Charge Distributions - http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Gauss/SpacialCharge.html

The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity Current density is a measure of the Density of flow of a conserved charge.

Dictionary

-noun

(physics) The amount of electric charge per unit volume of space or unit area of a surface.

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